Strange Universes: An Introduction to Non-Euclidean Geometries

If you draw two straight lines on a piece of paper, it’s not difficult to keep them from crossing. Imagine, however, that the lines extended in both directions off the page and without end. Do these hypothetical lines cross? Surprisingly, this mundane question goes to the heart of our modern conception of space. Your experience might suggest that the lines will cross unless they head off the edge of the page at exactly the same angle. In that case we call the lines parallel; and this is the answer Euclid asserts with his fifth (or “parallel”) postulate of the Elements. Roughly 2,000 years later, mathematicians came to the shocking realization that lines need not obey the parallel postulate. The resulting non-Euclidean geometries were so unexpected to the mathematicians who first conceived of them that one, János Bolyai, remarked, “Out of nothing I have created a strange new universe.” This course will explore the alternatives to Euclidean geometry that first appeared in the 18th century. These include hyperbolic, spherical, projective geometry, as well as more idiosyncratic geometries that we will devise together. Our exploration of these strange universes will be aided by visualizations that include drawing, computer graphics animation, and video game technology. Throughout, we will discuss the impact of the non-Euclidean revolution on astronomy, philosophy, and culture.